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Algebra Functions and Graphing
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Algebra Functions and Graphing

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Questions and Answers

If f(x) = 2x^2 + 3x - 1 and g(x) = x^2 - 2x, what is the value of (f - g)(2)?

  • 10
  • 14 (correct)
  • 12
  • 16
  • What is the range of the function f(x) = 2sin(x) + 1?

  • [0, 2]
  • [-2, 2]
  • [-3, 1]
  • [-1, 3] (correct)
  • If a circle has its center at (3, 4) and passes through the point (5, 6), what is its radius?

  • 5 (correct)
  • √10
  • 2
  • 3
  • What is the sum of the first 10 terms of the sequence an = 3n - 2?

    <p>150</p> Signup and view all the answers

    If f(x) = x^2 and g(x) = 2x + 1, what is the value of (f ∘ g)(1)?

    <p>6</p> Signup and view all the answers

    What is the equation of the line that passes through the points (2, 3) and (-1, 4)?

    <p>y = -x + 5</p> Signup and view all the answers

    If f(x) = sin(x) and g(x) = cos(x), what is the value of (f + g)'(π/4)?

    <p>-√2</p> Signup and view all the answers

    What is the equation of the circle with center (-2, 3) and radius 4?

    <p>(x + 2)^2 + (y - 3)^2 = 16</p> Signup and view all the answers

    If a series converges by the ratio test, what can be said about the absolute value of the common ratio?

    <p>it is less than 1</p> Signup and view all the answers

    What is the equation of the graph of f(x) = x^2, shifted 3 units to the right and 2 units down?

    <p>y = (x - 3)^2 - 2</p> Signup and view all the answers

    Study Notes

    Functions

    • Domain and Range:
      • Domain: set of all input values (x) for which the function is defined
      • Range: set of all output values (y) that the function can produce
    • Function Operations:
      • Sum/Difference: (f ± g)(x) = f(x) ± g(x)
      • Product: (fg)(x) = f(x) * g(x)
      • Composition: (f ∘ g)(x) = f(g(x))
    • Function Properties:
      • Even/Odd: f(-x) = f(x) or f(-x) = -f(x)
      • Increasing/Decreasing: f(x) ≥ f(y) or f(x) ≤ f(y) for x > y

    Graphing

    • Graph Types:
      • Linear: straight line, y = mx + b
      • Quadratic: parabola, y = ax^2 + bx + c
      • Exponential: y = a * b^x
      • Trigonometric: y = a * sin(x) or y = a * cos(x)
    • Graph Transformations:
      • Vertical Shift: y = f(x) ± k
      • Horizontal Shift: y = f(x ± h)
      • Reflection: y = -f(x) or y = f(-x)
    • Graphing Techniques:
      • Plotting points
      • Using symmetry
      • Identifying asymptotes

    Trigonometry

    • Angles and Triangles:
      • Degree measure: 0° ≤ θ ≤ 360°
      • Radian measure: 0 ≤ θ ≤ 2π
      • Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1
    • Trigonometric Functions:
      • Sine: sin(θ) = opposite side / hypotenuse
      • Cosine: cos(θ) = adjacent side / hypotenuse
      • Tangent: tan(θ) = opposite side / adjacent side
    • Trigonometric Identities:
      • Sum and Difference Formulas
      • Double and Half Angle Formulas

    Analytic Geometry

    • Coordinate Systems:
      • Cartesian Coordinates (x, y)
      • Polar Coordinates (r, θ)
    • Equations of Lines:
      • Slope-Intercept Form: y = mx + b
      • Point-Slope Form: y - y1 = m(x - x1)
      • Standard Form: Ax + By = C
    • Circles and Conic Sections:
      • Circle: (x - h)^2 + (y - k)^2 = r^2
      • Ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
      • Parabola: y = a(x - h)^2 + k

    Sequences and Series

    • Sequences:
      • Arithmetic Sequence: an = a1 + (n - 1)d
      • Geometric Sequence: an = a1 * r^(n - 1)
    • Series:
      • Arithmetic Series: Σan = (a1 + an)/2 * n
      • Geometric Series: Σan = a1 * (1 - r^n) / (1 - r)
    • Convergence Tests:
      • nth Term Test
      • Ratio Test
      • Root Test

    Functions

    • Domain of a function refers to the set of all input values (x) for which the function is defined.
    • Range of a function refers to the set of all output values (y) that the function can produce.
    • Function operations include sum/difference, product, and composition, with formulas (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x) * g(x), and (f ∘ g)(x) = f(g(x)) respectively.
    • Functions can have even or odd properties, where f(-x) = f(x) or f(-x) = -f(x) respectively.
    • Functions can also be increasing or decreasing, where f(x) ≥ f(y) or f(x) ≤ f(y) for x > y.

    Graphing

    • Linear graphs are straight lines represented by the equation y = mx + b.
    • Quadratic graphs are parabolas represented by the equation y = ax^2 + bx + c.
    • Exponential graphs are represented by the equation y = a * b^x.
    • Trigonometric graphs are represented by the equations y = a * sin(x) or y = a * cos(x).
    • Graph transformations can be vertical shifts (y = f(x) ± k), horizontal shifts (y = f(x ± h)), or reflections (y = -f(x) or y = f(-x)).
    • Graphing techniques include plotting points, using symmetry, and identifying asymptotes.

    Trigonometry

    • Angles can be measured in degrees (0° ≤ θ ≤ 360°) or radians (0 ≤ θ ≤ 2π).
    • The Pythagorean Identity is sin^2(θ) + cos^2(θ) = 1.
    • Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions, where sin(θ) = opposite side / hypotenuse, cos(θ) = adjacent side / hypotenuse, and tan(θ) = opposite side / adjacent side.
    • Trigonometric identities include sum and difference formulas, and double and half angle formulas.

    Analytic Geometry

    • Coordinate systems can be Cartesian (x, y) or polar (r, θ).
    • Equations of lines can be in slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), or standard form (Ax + By = C).
    • Circles can be represented by the equation (x - h)^2 + (y - k)^2 = r^2.
    • Ellipses and parabolas are types of conic sections, represented by the equations (x - h)^2/a^2 + (y - k)^2/b^2 = 1 and y = a(x - h)^2 + k respectively.

    Sequences and Series

    • Arithmetic sequences have the form an = a1 + (n - 1)d, while geometric sequences have the form an = a1 * r^(n - 1).
    • Arithmetic series have the formula Σan = (a1 + an)/2 * n, while geometric series have the formula Σan = a1 * (1 - r^n) / (1 - r).
    • Convergence tests for series include the nth term test, ratio test, and root test.

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